SRM phase and poles configurations.
Switched reluctance machines (SRM) are an alternative to conventional and permanent magnet (PM)-based machines. They are simple, robust and fault tolerant, and able to reach very high speeds with high efficiency. However, they operate with high torque pulsation and are noisy. Also, a nonconventional power converter type and specific control schemes must be included. Furthermore, SRM operation is characterized by high nonlinear features, which makes it difficult to be modeled and controlled. SRM energy conversion principles are a keystone to understand its operation. SRM efficiency increases with speed, where core and mechanical losses are more significant. For this machine, core loss estimation is a complex task, due to the nonlinear behavior of the magnetic materials. In addition, flux waveforms are not sinusoidal and particular waveforms appear in different core sections. Empirical formulas are usually considered in core loss estimation, but this is insufficient for SRM.
- SRM geometric and magnetic features
- energy balance
- torque production
- loss characterization
Switched reluctance machines (SRM) have been recognized as an alternative to conventional AC and DC motor drives. They have a multi-domain potential, such as electric vehicle, household appliances, aircraft industry, and servo system, to name a few examples . They are simple, rugged, and fault-tolerant machines, with low cost. Moreover, they can reach very high speeds with high efficiency, since there are no windings or permanent magnets (PM) in the rotor . A considerable attention to SRM drives has been given by both industry and research community, mainly due to the absence of PM. Limitations on rare-earth element availability and their increasing cost are the main reasons, since those elements are fundamental in PM composition . Nonetheless, there are some drawbacks that need to be tackled: they have high torque pulsation and noisy operation. Also, a nonconventional power converter type with specific control schemes must be included. Furthermore, its efficiency (low speed) and power density are lower than in PM machines [3, 4].
SRM modeling is fundamental for two basic reasons: design and control development. In the first one, reducing prototype costs, while optimizing its design (e.g., better efficiencies with reduced torque ripple and noise), is a crucial goal . On the other hand, coupling the SRM model with power converter and load models is fundamental for developing better control algorithms .
SRM operate under strong nonlinear conditions, where modeling can be a very challenging task. Finite element analysis (FEA) is a very accurate but time-consuming method. Analytical models and lookup tables are much faster but have reduced accuracy. Therefore, SRM modeling must be addressed as a trade-off between simulation accuracy and computational speed .
For any electrical machine, loss estimation is fundamental for design optimization and control. Furthermore, an accurate loss evaluation is vital for achieving the desired operation conditions . Like in other electric machines, SRM losses occur in stator windings, core materials (stator and rotor), and mechanical moving parts. In high-speed operation, where SRM has better performances, core losses have a significant weight, together with the mechanical ones. Iron loss estimation in SRM is a complex task, mainly due to nonlinearities of the magnetic core. In addition, flux density waveforms are not sinusoidal, and different ones must be considered in several magnetic circuit parts, depending on the machine geometry . Moreover, those waveforms are conditioned by control features.
This chapter is structured as follows: Section 2 starts with an overview on reluctance machines. Next, fundamental geometric and magnetic features, for conventional SRM, are addressed. Operation principles are discussed, highlighting the correlation between machine features and control parameters. In Section 3, energy conversion is deeply analyzed, based on a lumped parameter model. The focus is given to torque production mechanism. Section 4 addresses SRM losses, where common methods for SRM loss estimation are discussed. The impact of sinusoidal and nonsinusoidal flux density waveforms is also discussed.
2. SRM characterization
This section intends to present a general SRM overview. Several subjects are addressed, starting with the main advantages and drawbacks. Geometric and magnetic features, as well the operation principles, are also discussed.
2.1 Reluctance machines (RM) and SRM basic features
For every RM, torque production is a consequence of rotor movement to a position where the self-inductance of the excited winding is maximized (i.e., minimum magnetic reluctance). The motion may be rotary or linear, and the rotor may be interior or exterior . This is a different principle from conventional machines, such as induction, synchronous (cylindrical rotor), or DC motors, where torque development results from the combination of two magnetic fields (stator and rotor).
Stator and rotor magnetic circuits are made of laminated soft magnetic iron. Generally, windings are located only in stator, consisting in several electrically independent phase circuits. They may be excited separately (e.g., SRM and stepper motor) or together (e.g., synchronous reluctance motor, SRM, and stepper motor). The rotor has no windings or permanent magnets, and its shape has a significant role in maximizing the phase inductance variation with its position. In fact, all these machines have in common a magnetic circuit with deep anisotropic features. The rotor simplicity is the main advantage when compared to permanent magnet or rotor winding machines. The manufacturing cost can be lower than in other types of motors; also, the reliability and robustness are improved and rotor cooling is not so critical . Figure 1 depicts the basic structure of SRM and synchronous reluctance motor (Synch_RM):
SRM higher simplicity can be observed: stator winding coils are concentrated around its salient poles (Ns); usually, diametrically opposite windings are connected in series. Motor phases can be formed by one of these pairs or groups of pairs. The rotor geometry has also a set of salient poles (Nr)—in this case Ns = 6 (two poles/phase), while Nr = 4.
Synch_RM has an AC conventional slotted stator, where the windings are sinusoidally distributed. Figure 1(b) represents a four-pole rotor with axially magnetic laminations.
Although SRM presents a similar structure when compared to variable reluctance (VR) stepper motors, there are important differences that should be highlighted: usually SRM have a small pole number, with a larger stepping angle. Most often, the rotor has a continuous movement, and its output power is much higher than in VR stepper motor .
In conventional SRM design, stator and rotor poles features are usually attached to the following conditions:
where Ns and Nr are, respectively, the stator and rotor poles (even numbers); βs and βr are, respectively, the stator and rotor pole arcs; is the stroke angle, i.e., the rotor displacement due to a single pulse current; and is the SRM phase number.
The denominator in Eq. (3) gives the total phase pulses/revolution, while Eq. (4) assures the motor can start, independently of rotor initial position. For an m-phase SRM, there are several (Ns; Nr) combinations, according to Eqs. (1) and (5). Table 1 shows the most common configurations.
Usually, the average torque is higher for large Ns and Nr, with a smaller ripple. However, the number of power devices (switches and diodes) in the converter will increase. Therefore, the cost is higher, as well the switching losses. Conduction time is also higher, which increases copper losses. In addition, since phase current frequency (fs) is given by fs = ωr*Nr (ωr: rotor angular speed), core losses will be higher. Nevertheless, torque ripple filtering becomes easier, due to its higher frequency .
2.2 SRM background and development
The SRM drives for industrial applications are relatively recent. Although the earliest recorded SRM is from 1838 , its actual development stage would not be possible without the semiconductor-based solid-state power switching technology in the early 1960s. This brought a huge interest in new machine development, as well in control methods, where the SRM is included.
In the last 40 years, different configurations of reluctance motor and switching methods have been investigated, both theoretically and experimentally. This is quite clear when the 67 worldwide registered patents on SRM before 1976 are compared to the 1755 ones, between 1976 and 1999 . According to this reference, the total number of papers published in these periods increased from 11 to 1847, respectively. Presently, most significant SRM patents are the basis for its manufacture. The development of digital control and power electronics has made possible to exploit in an effective way the SRM characteristics .1
From the power conversion perspective, the SRM main disadvantage is related to the absence of permanent magnets and windings in the rotor: the machine power density is limited by the single stator excitation source . Also, due to the magnetic circuit geometry, the machine is very sensitive to the effects of fringing fields and magnetic saturation. This introduces high nonlinear features that must be considered in SRM operation. A power converter and a controller must be integrated with the SRM, since phase currents are electronically commutated. The consequent pulsed magnetic fields have a major contribution in torque ripple and acoustic noise, which are the main drawbacks of SRM drives . In order to tackle these drawbacks, research efforts have been focused on control strategies [13, 14], but machine design  and power converter structure  have also been addressed.
Another relevant SRM particularity is a small phase mutual inductance. In many applications it is considered null, i.e., each phase is magnetically independent from each other. This has a deep impact on the machine operation: since a phase fault has no influence in the other ones, SRM have a significant fault-tolerance capacity. As phase number increases, the smaller is the impact of a faulted phase. However, in high-speed operation, currents in adjacent phases can overlap for a significant portion of the conduction cycle. Therefore, the mutual flux linkages between phase windings can be most relevant, so their effects should be considered. This suggests that different model and control approaches must be addressed, whether SRM is operating in low or high speeds.
2.3 Geometric specifications
SRM phase inductance depends on both the current (due to saturation effect) and rotor position (). The unsaturated values of the aligned inductance (La) and the unaligned inductance (Lu) are fundamental in SRM basic theory, in particular the ratio La/Lu . In addition, the machine geometric features have also a significant role, particularly the size and shape of stator and rotor poles, as well the air gaps. Figure 2 depicts some of the most relevant geometric parameters, given by:
2.4 Magnetic features
Figure 3 represents a set of magnetization curves, for one phase of an SRM, where each one is related to a different rotor position (). The aligned and unaligned positions are the set limits. Also, the influence of magnetic saturation is visible.
Admitting uniform flux density, with no flux leakage and no saturation/hysteresis effect, the phase inductance can be approximated by:
where: μfe and μ0 are, respectively, the core material and the air magnetic permeability; Ap is the cross section of one stator and rotor magnetic pole (considering similar sections); Ay is the cross section of one yoke section (considering similar sections); lp is the phase linkage flux longitudinal length in stator pole + rotor pole path; l0 is the phase linkage flux longitudinal length in the air gap path; ly, is the phase linkage flux longitudinal length in yoke path between two stator magnetic poles.
Under the previous conditions, for both the aligned and unaligned position, one has A(θ) = Ap. The airgap term in Eq. (9) denominator is dominant, so l0 has a profound impact in La and Lu. Moreover, since SRM anisotropic features play a fundamental role on its performance, maximizing La/Lu is crucial in SRM design. In order to increase La, the correspondent l0 should be as small as possible. This highlights the fact that SRM aligned air gap must be tighter than for other machines. As a consequence, manufacture imperfection or rotor eccentricity will have a considerable negative impact in SRM operation .
From Figure 3, it can be seen that Ls() gradient is more relevant in the overlapping region, meaning that this is the most relevant torque production zone. In the overlapping regions, Ls varies approximately linearly with. Thus, it is reasonable to admit A() as:
machine’s axial length; R rotor pole radius; βr rotor pole arc.
Nonetheless, it should be noted that A() is in fact much dependent on the machine pole shape.
2.5 SRM operation fundamentals
Torque production conditions can be seen in Figure 4, where an elementary single-phase machine is depicted, for different rotor positions.
In (a), rotor poles are perfectly aligned with the stator phase poles, where its inductance is maximum. Thus, no torque is produced. The rotor unaligned position is depicted in (d): its interpolar axis is aligned with the stator poles. The torque has a null value, since the rotor poles are displaced in an equivalent position related to the stator poles (here the phase inductance is minimum, since magnetic reluctance is maximum). As the rotor moves from the unaligned position and comes closer to any stator pole, the torque is no longer zero and tries to align the rotor with that particular stator pole: in (b) and (c), the developed torque tends to bring the rotor back to (a) position—this is a stable equilibrium position, while (d) is an unstable equilibrium position . Torque direction is contrary in (b) and (c), which highlights the fact that switching currents in stator windings must be implemented according to the rotor position—it becomes clear that stator phases must be energized for rotor position between maximum and minimum phase inductance. This elementary machine highlights three important SRM general features [9, 10]:
In order to avoid situations (a) and (d) in Figure 4, all SRM verify the condition: Ns ≠ Nr. In those positions the motor is not able to start when stator phase is excited.
Torque is produced by sequentially switching the currents in the stator phase windings when there is a variation of reluctance, which depends on rotor position. This is the reason why a controlled power converter supply system is mandatory with rotor position feedback. Therefore, SRM operation is always implemented with a converter + controller (SRM drive).
Winding current polarity has no influence in torque development, since this is a reluctance torque.
The fundamental concepts in correlating SRM rotor position to its control are depicted in Figure 5. Phase current (I()) is represented for both motor and generator SRM modes, as well the torque production zones (T()). Except for the red ) curves which denote saturation effect, all the others assume ideal conditions (no saturation and hysteresis; ohmic winding resistance is zero).
It can be seen that Ls is a periodic function of , with period .2 The following expressions can be considered:
The instantaneous torque is controlled by current amplitude regulation and the switching instants (). In addition, rotor speed has also a significant impact on control modes.
The ) top flat zone is related to the difference between the stator and rotor pole arcs, respectively, βs and βr: as it tends to zero, the top flat becomes narrower. Although this is a “dead zone” since there is no energy conversion (torque is null), it might help to avoid braking torque in the demagnetizing period (TOFF). It should be emphasized the unidirectional current polarity in both modes. Torque development is clearly associated to (i.e., the overlapping poles regions); it is associated to motor (+) and generator (−) modes. This stresses the fact that current impulse rise (TON) and fall (TOFF) periods have a fundamental role in SRM operation features: TON must be as small as possible, while current should reach its maximum when stator and rotor poles start to overlap. As for TOFF (the defluxing period), it depends on how fast the energy stored in the linkage magnetic field can be turned to zero. Therefore, TOFF limits must assure a zero current at the beginning of the , in order to avoid a braking torque. These are important issues that must be addressed by the drive controller.
The SRM core magnetic features, together with pole geometry, have a fundamental role in the Ls(θ) profile. This is crucial in estimating and values. Moreover, magnetic characteristics are particularly influenced by local saturation in stator and rotor poles, which also depends on SRM operation conditions.
Phase current commutation, in addition to SRM geometry, is responsible for torque ripple components. This contributes for increasing the complexity demanded to the SRM drive controller.
In short, the phase current pulse waveform as a function of , in addition to and , plays a keystone role in all SRM operation modes. Therefore, the machine must be integrated with the power converter and the controller, working as a single system.
3. SRM modeling and energy balance
SRM can be modeled in a similar way as for conventional electrical machines. Therefore, a model based on lumped parameters can be a first option. However, depending on the purpose (e.g., design or control), particular attention should be addressed when applying conventional methods for modeling this machine, due to its specific nonlinear operation features.
Nonetheless, this section intends to address the SRM energy conversion principles, where a lumped parameter approach is well suited. The mathematical model is based on the following equations:
where Rs, i, and Ψ(i,θ) are, respectively, the stator phase resistance and current and flux linkage.3
where Ls(i,θ) is the stator phase self-inductance.
Electromechanical equation (instantaneous torque)
where is the torque developed by phase j and m is the SRM phase number.
where Tload is the mechanical load, J is the system inertia, is the instantaneous rotor mechanical angular speed, and Kf is the viscous friction coefficient, due to bearing lubricant fluid.
It should be noted that magnetic feature complexity is not entirely depicted in Figure 3 (flux density waveforms in different SRM sections cannot be addressed). The high complexity of SRM modeling lies in Eq. (17), which has a significant impact on torque production, since it depends on the magnetic anisotropic features. In order to take a deeper insight on Eq. (18), the torque production mechanism is now analyzed with more detail.
3.1 Electromechanical energy conversion and torque production
Since SRM are electromechanical energy converters, a review on energy balance relationships is first addressed. An electromechanical system is based on an electrical system, a mechanical system, and a coupling field (i.e., electromagnetic and electrostatic fields), common to both systems. From this interaction, energy can be transferred from one system to the other. It should be noted that both coupling fields can exist simultaneously, while the electromechanical system may have several electrical and mechanical subsystems. Different losses by heat dissipation occur in the electrical and mechanical systems, as well in the coupling field. Several phenomena contribute to this :
Electrical losses, due to current-carrying conductor ohmic resistance
Mechanical losses, due to friction of moving parts
Magnetic losses in the core ferromagnetic materials (magnetic fields) and dielectric losses (electric fields)
Figure 6 depicts the energy balance in an elementary electromechanical system—according to the energy conservation principle (first thermodynamic law)—where losses and stored energy components are represented.
The black dotted arrows represent the adopted convention for positive W’s. The energy balances for the electric and mechanical systems are then expressed as:
where WE is the energy supplied by the electrical source, WeL is the electrical losses, WeS is the energy stored in the fields (electric or magnetic) not coupled with the mechanical system, and Wef is the energy transferred to the coupling field from the electrical system. In a similar way, WM is the energy supplied by the mechanical source, WmL is the mechanical losses, WmS is the energy stored in the moving parts of the mechanical system, and Wmf is the energy transferred to the coupling field from the mechanical system. The colored arrows represent the physical energy flux: except for the losses (irreversible phenomena), all the W’s flux can be reversible, depending on the system operation. From Figure 6 it can be seen that:
From this point, only electromagnetic coupling field are considered. Rearranging Eq. (23) gives:
The magnetic coupling field will be addressed as a conservative field, in order to take advantage of a fundamental property: the energy stored in a conservative field depends only on state variable values, and not on the transition between states. The sum of potential and kinetic energy is constant; therefore, the losses are null (i.e., ). It should be noted that, for the purpose of this analysis, such approach is quite acceptable, since the most significant part of the stored energy in the coupling fields is in the air gaps of the electromechanical device. Air is a conservative medium; thus the energy stored there can be returned to the electrical or mechanical systems. Finally, Eq. (24) can be written as:
The voltage equation for a generic electric system (one phase) is given by Eq. (16). For convenience of analysis, it is written again as:
Therefore, Eq. (25) can be expressed as:
Since is the flux linkage, the effect of leakage inductance is neglected. Therefore,and the following differential equation can be extracted:
It is important to stress that is a single-value function, since it represents a conservative field. Therefore, hysteresis is not included here, but saturation can be considered—this is the reason for having “i” as an independent variable. With as the other independent variable, one has:
Substituting this in Eq. (33) yields:
The function allows a general formulation for infinitesimal variations in the coupling field stored energy and the mechanical energy. However, analytical expressions for are difficult to get. At this point, it is convenient to define the coenergy function (Wc) as:
Under this condition, is given by:
Both and are represented in Figure 7 (i: 0 → i1, = constant).
From Figure 7, it can be seen that (for i = constant).
Taking Eq. (39) θ derivative, one has:
In order to take an insight over the torque production in an SRM, the analysis is now addressed taking di = 0. Under this condition, Eq. (36) is written as:
For a generic SRM with m phases, the total electromagnetic torque is:
where Wjc is the coenergy related to the stored magnetic field of phase j and m is the SRM phase number. This highlights the fact that torque production is in close relation with the core magnetic properties, for each rotor position (). The average torque (Tav) can be calculated by integrating :
A geometric representation may help to clarify how energy is transferred between the electrical domain, the magnetic coupling field, and the mechanical domain. Figure 8 includes two magnetic characteristics for two rotor positions, θa and θb.
For i =, is the energy transferred to the magnetic coupling field from the electrical source, for a rotor position displacement from θa to θb (A → B in Figure 8(a). This is related to the difference between rectangles [0--B-] and [0--A-] (i.e., the blue rectangle). From Figure 8(b), it can be seen that this difference is given by:
(and are the coenergy (Wc) related to, respectively, θa and θb.)
For an infinitesimal energy change (:
which is similar to Eq. (41).
3.2 Linear magnetic circuit
If no magnetic saturation exists, the magnetization curves are straight lines (for a fixed ), as represented in Figure 9.
Since , from Eq. (35), one has:
From Figure 9, it can be seen that:
In other words, for i =, the energy transferred to the magnetic coupling field from the electrical source, as a result of a rotor displacement, is evenly distributed by the stored energy variation in the coupling field and by the coenergy variation (i.e., mechanical energy variation). Finally, Eq. (51) is written as:
From Eq. (34), the instantaneous torque for a rotor position is given by:
Some Section 2.5 observations are now more clear from this linear mathematic torque formulation:
Phase current polarity has no influence in torque production. This is an important feature, since phase current pulses (and flux linkage) can be unipolar. In addition to a simpler controller (compared to other machines drives), iron losses are also smaller.
Torque development is associated to the growth (+) or decline (−) of stator phase inductance curve (), respectively, for motor and generator modes.
4. SRM loss characterization
As for any electric machine, SRM models must include losses, since its performance is quite dependable on it. Moreover, all losses should be accurately estimated for different operation modes . Compared to conventional AC machines, this is the most demanding task, particularly for the core losses, because flux density waveforms are not sinusoidal . Similar to other machines, losses in an SRM are related to copper, iron core, and mechanical ones. The machine geometry and operation conditions make SRM loss characterization a nontrivial task.
4.1 Overview on developed SRM loss estimation
Different methods for core loss evaluation in SRM have been investigated by several authors. Using finite elements methods, flux density waveforms are obtained by simulation in . The iron losses were then calculated based on the modified Steinmetz equation. In  an analytical method for calculating SRM core losses is proposed. The machine magnetization curves are extracted through finite element simulation, considering different control strategies. The flux density waveforms in the different parts of the SRM are derived from the flux density waveform of the stator pole, obtained by simulation. The specific core losses are separated into hysteresis, classical eddy current, and excess losses. Their calculation is based on the waveforms and time derivatives of the flux density in each zone. In , SRM efficiency is evaluated for continuous operation mode (i.e., current does not extinguish during each phase excitation), but only core losses are considered with more detail (copper losses are calculated through the current rms value and the DC phase resistance). Fourier flux density decomposition in the machine core sections is then applied, where core losses are calculated with the modified Steinmetz equation. In order to consider the minor loop influence, a correction factor is included. In  a SRM efficiency map is derived based on Fourier flux density decomposition. However, only copper and hysteresis losses are considered. In  core losses were calculated through simulation, based on analytic magnetic circuits, considering six different core materials. This approach was validated by comparing the simulation results to experimental ones. However, the paper does not provide information about the magnetic circuit approach.
In the following, an overview on iron loss estimation is addressed, from an engineering perspective. The evolution of the most relevant methods, as well as their merits and limitations, is discussed. Except for , the approaches mentioned in the previous paragraph are included here. In the last decades, considerable efforts have been addressing these issues. Nonetheless, further improvements are needed for a precise determination of the flux waveforms and correct calculation of iron loss at nonsinusoidal excitation and nonuniform flux distribution, for a wide range of operation scenarios. This is still an open research field.
As already stated, the focus will be given to iron loss. However, copper losses are usually more significant. Therefore, they are first addressed.
4.2 Copper losses
In the presence of nonuniform electric field (E) and current density (J) distributions, the power dissipated in an arbitrary volume (V) of a conductor is given by:
The vector (Ohm’s law) is expressed as4 (vectors E, J):
Thus, may be formulated as follows:
where ρ is the conductor material resistivity (Ω·m). For uniform E and J distributions, Eq. (57) can be written as:
I = J·A and RDC = ρ·l/A (RDC is the ohmic resistance of the conductor volume (V), where l is the length of the current (I) path inside V and A is the cross-sectional area.
An electrical machine winding is made of several coils, where different phenomena may have an impact on winding losses: ρ increases with temperature (temp), while the skin (kskin) and proximity (kprox) effects have a relevant impact on J distributions, particularly in high frequencies. All these effects should be included for an accurate winding loss characterization, meaning that Eq. (57) must be considered. For a lumped parameter model approach, as in Section 3, the previous effects are addressed by:
where =(temp, kskin, kprox).
Therefore, SRM total copper losses can be calculated based on Eq. (59), as the sum of all phase winding losses. Since phase current waveforms are periodic and not sinusoidal (with high frequency), Fourier series may be applied in order to get its harmonic components . It follows that the total winding losses are calculated as:
where m is the phase number and RAC_i and Ii_rms are, respectively, the phase AC resistance for the i-order harmonic and the current rms of the i-order harmonic.
In short, copper loss estimation is a relatively simple task, using the stator current. However, the skin and proximity effects require special attention, particularly for high-frequency components. Furthermore, for similar reasons discussed in the next section, Fourier analysis should be carefully applied. Particularly for high loads, temperature influence must also be included.
4.3 Core losses
Usually, lamination manufacturers provide core loss data under sinusoidal excitations in a limited frequency and flux density range. However, this kind of data is not adequate for predicting losses in electrical machines with nonsinusoidal flux waveforms. This requires loss information at high frequencies and high flux densities, particularly for high-speed operation, as in SRM .
The development of nonsinusoidal flux density machines and conventional machines fed by power converters (e.g., SR, brushless DC machines, and induction motors with PWM voltages) has motivated researcher’s efforts to reach a deeper characterization of magnetic losses under such excitation waveforms. This can be done either by measurement or estimation [19, 24]. New challenges arise for both approaches, since sinusoidal-conventional methods are clearly insufficient. Moreover, since different flux density waveforms are related to particular electric machine configurations, this suggests that specific approaches for characterizing core losses must be addressed, according to the machine type .
SRM core loss modeling is a challenging task, since the nonsinusoidal flux density waveforms depend on motor design (geometry and number of stator and rotor poles, yoke geometry, number of phases), operating conditions (conduction angle and mechanical speed influence the magnetic saturation and lamination skin effect level), and the type of control used . Core losses are more significant as speed increases, so for applications like hybrid and electric vehicles, they must be carefully addressed.
Different approaches have been proposed for calculating SRM core losses, from theoretical supported empirical models5 to physical mathematical models . The later are more accurate, since they take an insight on the core loss physics, aiming to describe its nonlinear mechanism under distorted flux . However, they are complex and computationally heavy. From an engineering perspective, empirical models are the preferred ones, due to its simplicity and faster processing. Parameter estimation is based on curve-fitting methods, validated by manufacturer iron sheet data and experimental results or through finite element modeling [20, 24]. Accuracy is much sensitive to parameter values, so their estimation must be attached to specific conditions (e.g., flux density and frequency ranges). Moreover, the manufacturing process of the machine has also a deep impact on core losses; this is very difficult to address in the parameter estimation process [25, 26]. Cutting and punching operations have a relevant influence in the material properties, since they can create inhomogeneous stresses inside the sheets. This depends on the alloy composite, whereas the grain size in the sheets seems to be the main influencing factor .
4.3.1 Core loss estimation
Electrical machine core losses can be addressed by three different approaches, in time or frequency domain: empirical equations, loss separation components, and hysteresis models. Only the first two are discussed here.
18.104.22.168 Empirical models based on Steinmetz equation
The Steinmetz coefficients depend on both frequency and flux density , so in a waveform with relevant harmonics, it might be difficult to find their values. Based on results of many tests, the classical Steinmetz equation was the first attempt to calculate core loss :
where Bmax is the peak value of the flux density at the lamination, f = 1/T is the remagnetization frequency (T is the hysteretic cycle time interval), and coefficients Cm, α, and β are estimated by fitting the loss model to the lamination manufacturer or measured data. It must be pointed that Eq. (61) assumes sinusoidal flux densities, with uniform distribution across the lamination thickness. Over the years, several upgrades were performed, in order to account for nonsinusoidal waveforms.
The modified Steinmetz equation (MSE) aims to calculate core loss under arbitrary B waveforms. The macroscopic remagnetization rate dM/dt (which is proportional to dB/dt) is directly related to the core losses, as a consequence of wall domain motion . Eq. (61) is then replaced by:
where ΔB = Bmax-Bmin.
The MSE has the advantage to highlight the physical origin of the losses, with the same parameters as in Eq. (61). A disadvantage is that it underestimates losses for waveforms with a small fundamental frequency part. Another difficulty is the treatment of waveforms with multiple peaks, in which peak-to-peak amplitude is not an enough description . In fact, this reflects the increasing MSE limitations, as the flux density moves away from the pure sinusoidal waveform.
The generalized Steinmetz equation (GSE) was also developed from the original Steinmetz equation , where losses are calculated in time domain. Here, the instantaneous iron loss is assumed to be a single-valued function of the rate dB/dt and B(t). The inclusion of B(t) allows to consider the DC bias influence in the loss calculation, without additional measurements or curve-fitting functions:
For different frequency ranges, different parameter values are necessary. Thus, in the presence of relevant harmonics, the accuracy decays. This is an important drawback, particularly in the presence of minor loops. To overcome this limitation, in  the flux density waveform is split into a major loop and minor loop(s), in order to consider the later one effect over loss calculation. Nonetheless, DC bias influence is no longer included. This approach was named improved GSE (iGSE). In  a test for loss calculation in nanocrystalline materials, with nonsinusoidal excitation voltage, is reported. Harmonic decomposition was considered, including the phase displacements. According to this reference, iGSE gives better results than GSE.
Both MSE and GSE are applied in time domain. It must be pointed out that the time evolution (history) of the flux density waveform is neglected. This has an impact on the physical phenomena insight, where loss evaluation may be affected.
22.214.171.124 Loss separation method
For a general scenario, the hysteresis loss density related to one cycle, in a particular core part, can be calculated by the following expression:
where B and H are, respectively, the magnetic flux density and the magnetic flux strength, f is the cycle frequency, and is the density of the ferromagnetic material6. The eddy current loss density is derived from Maxwell equations, which is given by:
where is the material electric conductivity7. However, in many situations, these equations are unpractical, even with finite element analysis. The complex nonlinear B(H) characteristic, which is also dependent on the lamination thickness, is the main reason . So, empirical models for core loss evaluation are often considered. The most common is the Steinmetz equation : the following expression for core loss density in a ferromagnetic material (formulated as the sum of hysteresis () and classic eddy current () losses) was first achieved:
which is valid in the range of 0.1 T < Bmax < 1.5 T. Hysteresis and eddy current loss coefficients are, respectively, and , which can be extracted from measured data. They both depend on the core material; also depends on the lamination thickness (d). In fact, has an analytical formulation, which can be derived from Eq. (66):
An important remark is that Eq. (65) assumes uniform flux density distribution across the lamination thickness. It should be noted that for SRM this is a most relevant limitation. Over the years, Eq. (67) has been upgraded, which brought higher accuracy. Experimental data showed that the measured eddy current losses are higher than . Based on statistical loss theory, Bertotti proposed an additional term to account for the excess losses (), which can be expressed as :
where is dependent on the material microstructure, the conductivity, and the cross-sectional area of the lamination. Different theories have been developed to explain excess losses, but this is still under discussion .
For the hysteresis term, the power of Bmax was found later to be dependent on the material type, as well on Bmax. Therefore, a more accurate expression was adopted for core loss estimation, with a modified hysteresis term (, a, and b are its parameters) and including the excess losses:
Further experiments, together with finite element analysis, revealed that it predicts core losses with very good accuracy for f ≤ 1500 Hz and Bmax < 1 T. For higher flux densities, it gives good results for frequencies up to 400 Hz .
126.96.36.199 Core losses under nonsinusoidal and uniform flux densities
Most often, coefficients in Eq. (70) are obtained through curve-fitting techniques, based on manufacturer’s core losses. Moreover, the hysteresis term in Eq. (70) is limited to symmetrically flux density variations about zero (i.e., −Bmax < B < Bmax) and, most important, without minor loops included in the main hysteresis loop . In short, the previous expression is mostly suited for sinusoidal flux density waveforms, with uniform distributions in the core. For SRM, these conditions are not valid, so different approaches must be considered. Furthermore, parameter for empirical formulas related to nonsinusoidal flux losses must be based on measured core loss values. Manufacturer’s data for sinusoidal flux may lead to large errors for nonsinusoidal core loss estimation. In order to consider nonsinusoidal waveforms, the product () is substituted by (dB/dt) in these two terms:
where T = 1/f.
However, since SRM usually operates at high frequencies, the influence of skin effect and saturation over eddy current and hysteresis losses cannot be disregarded: it leads to nonuniform field densities, which brings additional challenges for core loss modeling. In order to have accurate estimations, this must be addressed. In the following, a brief discussion about the impact on eddy current and hysteresis losses is addressed.
188.8.131.52 Eddy current losses
The skin effect in the magnetic core is due to the field created by eddy currents: for high frequencies, particularly for thick lamination core, the flux density at the lamination surface is higher than the one at its center; moreover, their waveforms are also displaced [24, 32]. Flux penetration in the lamination is measured by the skin depth penetration (δ): it is the distance from the steel surface, where the field has decreased by a factor of 1/e, and it is approximately given by:
where f is the excitation frequency, the magnetic permeability, and is the material electric conductivity.
Therefore, the magnetic density field has a nonuniform distribution, and it is mainly concentrated on the lamination surface. Eddy current paths have now a higher resistance, meaning that its value and the correspondent losses are smaller, when compared to uniform field densities scenarios. Naturally, this is not foreseen by the previous expressions, which give excessive values. Several models to calculate eddy current loss in electrical machine laminations for nonuniform field density have been proposed . A common approach for a periodic nonsinusoidal flux density waveform is to consider its Fourier series decomposition. The contribution of each harmonic component is calculated based on its frequency and magnitude, through Steinmetz expression. Frequency is particularly relevant, since it determines the skin effect magnitude of the individual harmonics. However, one must not forget that such an approach is based on the superposition principle. In this way, its effectiveness must be always confronted with experimental loss values.
184.108.40.206 Hysteresis losses
Due to SRM geometry and operating conditions (in particular, high frequencies, local saturation, and skin effect), the peak flux density may be very different in several parts of the lamination . This causes local hysteresis loops, i.e., minor loops, in addition to the major loop (Figure 10).
As a consequence, several points inside the lamination have different local hysteresis power losses [24, 34], which may lead to core hot spots. It should be noted that a relevant harmonic content in the flux density waveform reflects a significant number of minor loops. Moreover, these additional losses may represent an important proportion of the total hysteresis loss, which stresses the importance of modeling them . A most relevant conclusion in this reference is that minor loop positions inside the main cycle (i.e., the DC flux density value associated to the minor loop) have a significant influence in the hysteresis losses. It should be noted that none of this is considered in the first term of Eq. (71).
Based on experimental studies, minor loop hysteresis loss evaluation has been frequently addressed through empirical formulas [20, 24] (in  the Fourier series harmonic decomposition is considered). However, one must be aware that the effectiveness of this approach is attached to certain simplifications and/or to specific flux waveforms. This highlights the fact that for every kind of electrical machine, under specific operation conditions, a particular formula should be addressed. Therefore, a lot of work still need to be done, in order to get accurate methods to estimate hysteresis minor loop losses . These are important issues to be addressed in SRM core loss characterization.
220.127.116.11 Rotational flux losses
Rotational flux densities (due to changes in the flux density vector direction, relatively to a given reference frame) may have important contributions for the total core losses in electrical machines.
Particularly in SRM, such rotational flux density variations are well pronounced around the stator and rotor tooth, as represented in Figure 11, due to changes on their relative position.
Predicting rotational flux losses is much more complex than alternating flux, with a fixed magnetic axis. Moreover, measuring them is quite complex, since precisely controlled rotational flux density waveforms are difficult to induce in test samples .
Some models have been proposed for estimating iron loss under rotational conditions but are based on sinusoidal flux densities. Usually, the flux density vector is decoupled into two orthogonal components, for a given machine region. For each component, losses are independently calculated and then added . Once again, this approach relies on the superposition principle. Due to hysteresis high nonlinearity, it seems reasonable to question the effectiveness of such approach.
SRM are simple, rugged, and fault-tolerant machines, with potential to compete with conventional motors. The major drawbacks are torque pulsation and acoustic noise, due to its high nonlinear operation features. In addition, the power converter requires a particular architecture, together with specific control methods. SRM modeling is a challenging task, due to its geometry and operation conditions. Nonetheless, this is a crucial task, since nowadays design and control developments are much dependent on simulation models. Independently of the modeling approach, a key issue is to understand the SRM electromechanical conversion process.
Accurate SRM loss characterization is crucial for designing and control algorithm development. Electrical machines core loss estimation has been supported by empirical models for sinusoidal waveforms. However, this is not suitable for SRM, since flux density waveforms are not sinusoidal and differ from one another, according to the core sections. Moreover, complex magnetic phenomena must be addressed, which are not considered in most empirical models.
This chapter has two main intentions. The first is to present a general view on conventional SRM, where a substantial effort is given on analytical description of electromechanical conversion. The second one is to give an overview on SRM losses—particularly, the iron loss—including the main challenges for characterizing and estimating them.
As a final remark, there are several open challenges in both subjects. A lot of research work still need to be done.
- Chapter 2 of this reference includes an interesting and detailed overview on SRM historic development.
- Ls(θ) origin is at the unaligned rotor position.
- Phase mutual influences and leakage flux are neglected.
- Valid for homogeneous and isotropic materials.
- In literature, they are referred as phenomenological models.
- For f [Hz] andmv [kg/m3], then Ph [W/kg].
- For E [V/m] andσ [S/m], then Pe [W/kg].